Price predictability at ultra-high frequency: Entropy-based randomness test

We use the statistical properties of Shannon entropy estimator and Kullback–Leibler divergence to study the predictability of ultra-high frequency financial data. We develop a statistical test for the predictability of a sequence based on empirical frequencies. We show that the degree of randomness grows with the increase of aggregation level in transaction time. We also find that predictable days are usually characterized by high trading activity, i.e., days with unusually high trading volumes and the number of price changes. We find a group of stocks for which predictability is caused by a frequent change of price direction. We study stylized facts that cause price predictability such as persistence of order signs, autocorrelation of returns, and volatility clustering. We perform multiple testing for sub-intervals of days to identify whether there is predictability at a specific time period during the day. • We present a test for predictability based on empirical frequencies. • The test statistics has a chi-squared distribution with known degrees of freedom. • We analyze properties of predictable financial time series at ultra-high frequency. • We study the degree of price predictability at different aggregation levels. • Predictable prices of some stocks exhibit a high probability of changing direction.